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July 1998 Nearest-neighbor walks with low predictability profile and percolation in $2+\epsilon$ dimensions
Olle Häggström, Elchanan Mossel
Ann. Probab. 26(3): 1212-1231 (July 1998). DOI: 10.1214/aop/1022855750


A few years ago, Grimmett, Kesten and Zhang proved that for supercritical bond percolation on $\mathbf{Z}^3$, simple random walk on the infinite cluster is a.s. transient. We generalize this result to a class of wedges in $\mathbf{Z}^3$ including, for any $\varepsilon \epsilon (0, 1)$, the wedge $\mathscr{W}_{\varepsilon} = {(x, y, z) \epsilon \mathbf{Z}^3: x \geq 0, |z| \leq x^{\varepsilon}}$ which can be thought of as representing a $(2 + \varepsilon)$-dimensional lattice. Our proof builds on recent work of Benjamini, Pemantle and Peres, and involves the construction of finite-energy flows using nearest-neighbor walks on Z with low predictability profile. Along the way, we obtain some new results on attainable decay rates for predictability profiles of nearest-neighbor walks.


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Olle Häggström. Elchanan Mossel. "Nearest-neighbor walks with low predictability profile and percolation in $2+\epsilon$ dimensions." Ann. Probab. 26 (3) 1212 - 1231, July 1998.


Published: July 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0937.60071
MathSciNet: MR1640343
Digital Object Identifier: 10.1214/aop/1022855750

Primary: 60J45 , 60K35
Secondary: 60J15

Keywords: Ising model , percolation , predictability profile , Random walk , transience

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.26 • No. 3 • July 1998
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